A332834 Number of compositions of n that are neither weakly increasing nor weakly decreasing.

0, 0, 0, 0, 1, 4, 14, 36, 88, 199, 432, 914, 1900, 3896, 7926, 16036, 32311, 64944, 130308, 261166, 523040, 1046996, 2095152, 4191796, 8385466, 16773303, 33549564, 67102848, 134210298, 268426328, 536859712, 1073728142, 2147466956, 4294947014, 8589909976

Offset

0,6

Comments

A composition of n is a finite sequence of positive integers summing to n.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Eric Weisstein's World of Mathematics, Unimodal Sequence.

Formula

a(n) = 2^(n - 1) - 2 * A000041(n) + A000005(n).

Example

The a(4) = 1 through a(6) = 14 compositions:
  (121)  (131)   (132)
         (212)   (141)
         (1121)  (213)
         (1211)  (231)
                 (312)
                 (1131)
                 (1212)
                 (1221)
                 (1311)
                 (2112)
                 (2121)
                 (11121)
                 (11211)
                 (12111)

Mathematica

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], !Or[LessEqual@@#, GreaterEqual@@#]&]], {n, 0, 10}]

Prog

(PARI) a(n)={if(n==0, 0, 2^(n-1) - 2*numbpart(n) + numdiv(n))} \\ Andrew Howroyd, Dec 30 2020

Crossrefs

The version for unsorted prime signature is A332831.

The version for run-lengths of compositions is A332833.

The complement appears to be counted by A329398.

Unimodal compositions are A001523.

Compositions that are not unimodal are A115981.

Partitions with weakly increasing or decreasing run-lengths are A332745.

Compositions with weakly increasing or decreasing run-lengths are A332835.

Compositions with weakly increasing run-lengths are A332836.

Compositions that are neither unimodal nor is their negation are A332870.

Cf. A007052, A072704, A107429, A328509, A329744, A332281, A332284, A332578, A332640, A332641, A332643, A332669, A332746.

Sequence in context: A193522 A187091 A034528 * A128758 A341384 A258343

Adjacent sequences: A332831 A332832 A332833 * A332835 A332836 A332837

Keyword

nonn

Author

Gus Wiseman, Feb 29 2020

Status

approved